integer programming
در نشریات گروه صنایع-
International Journal of Supply and Operations Management, Volume:10 Issue: 2, Spring 2023, PP 174 -186The scheduling of sports events has a high degree of complexity due to the large number, diversity, and the interdependence of events as well as the existence of conflicting objectives. The present study investigates the integrated scheduling of multiple types of sports events simultaneously. A variety of sports events, including competitions, management meetings, training camps, and training workshops, are scheduled simultaneously regarding the appropriateness of the timeslot for each event and the suitability of the length of the time interval between each pair of events. Previous studies were limited to detailed scheduling of single types of events. For instance, they determine when and where each team plays with other teams. However, the current paper takes a macroscopic view and sets the timeslots in which the competitions and other types of events will be held. The part of output which sets the competition timeslots can be further given as input to one of the existing algorithms which determine the detailed schedule of competitions. An integer programming formulation is developed. Numerical examples demonstrate that the general-purpose solvers cannot obtain the optimal solution of real-sized problems within an acceptable time. To solve larger problems, the fix-and-optimize matheuristic approach is employed. Numerical results show the satisfactory performance of this approach. To validate the proposed method, the sport events of the Karate Federation of Iran for a whole year are scheduled as a real case study. Finally, using the data of this case study, a sensitivity analysis is performed for some of the parameters.Keywords: Sports scheduling, Multiple sport events, Integer programming, Fix-and-optimize matheuristic, Case Study, Sensitivity Analysis
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هدف:
درصد قابل توجهی از دارایی ها در موجودی های انبار انباشته شده است، از این رو انبار در اقتصاد کشورها اهمیت فراوانی دارد. طراحی و چیدمان مناسب یک انبار، نقش بسیار زیادی در کاهش هزینه ها، کاهش زمان تدارک و تحویل، ارتقای بهره برداری از منابع و ارتقای سرویس دهی به مشتریان دارد. یک نوع از انبارهایی که اخیرا کاربرد زیادی پیدا کرده اند، انبارهای عبوری هستند که از نظر میزان کالاهایی که انبار می شوند و همچنین مدت زمان نگهداری آن ها، با انبارهای سنتی تفاوت دارند. هدف اصلی در این تحقیق مدل سازی و حل مسیله ای سازگار با شرایط دنیای واقعی است که کمتر موردتوجه پژوهشگران قرار گرفته است.
روش شناسی پژوهش:
برای حل مسیله مورد نظر، الگوریتم بهینه سازی گرگ خاکستری چند هدفه (MOGWO) بکار می رود و تنظیم پارامترها به کمک روش تاگوچی انجام می گردد.
یافته هابه کمک شاخص های متوسط فاصله از ایده آل، فاصله، تعداد نقاط پارتو و پراکندگی، توسط نمودار نسبت سیگنال به نویز، بهترین سطح ممکن برای پارامتر های الگوریتم تعیین شده است. با حل 10 مثال در ابعاد مختلف و بررسی نتایج و زمان حل آن ها مشخص شده است که با افزایش ابعاد مسیله، زمان حل نیز افزایش می یابد.
اصالت/ارزش افزوده علمی:
در این مطالعه، علاوه بر توجه به میزان مسافت طی شده در انبار که بیشتر مطالعات صورت گرفته در حوزه طراحی انبار بدان پرداخته اند، بهره گیری بیشتر از فضای قابل دسترسی در انبار و رضایت خرده فروشان نیز مد نظر قرار گرفته است. به همین منظور، یک مدل برنامه ریزی عدد صحیح مختلط، برای طراحی و چیدمان یک انبار عبوری در جهت حداقل نمودن مسافت ها، حداقل نمودن فضای خالی کف انبار و به حداکثر رساندن رضایت خرده فروشان پیشنهاد می شود.
کلید واژگان: الگوریتم بهینه سازی گرگ خاکستری چند هدفه، انبار عبوری، برنامه ریزی عدد صحیح، تاگوچی، طراحی و چیدمانPurposeWarehousing is very important in the economies of countries, because a significant percentage of assets are stored in the warehouse. Proper warehouse design and layout has a great role in reducing costs, reducing lead time and delivery time, improving resource utilization and customer service. One type of warehouse that has become widely used recently is cross dock warehouses, which differ from traditional warehouses in terms of the number of products stored and their storage time. The main purpose of this research is modeling and solving a problem that is compatible with real world conditions that have received less attention from researchers.
MethodologyMulti-Objective Gray Wolf Optimization (MOGWO) algorithm is used to solve the problem and Parameters are adjusted using the Taguchi method.
FindingsUsing the mean ideal distance, spacing, number of pareto solutions and diversification metric, the best possible level for the algorithm parameters is determined by the signal-to-noise ratio diagram. By solving 10 examples in different sizes and reviewing the results and their solution time, it has been determined that with increasing the size of the problem, the solution time also increases.
Originality/Value:
In this study, in addition to considering the distance traveled in the warehouse, which most studies have done in the field of warehouse design, more use of available space in the warehouse and the satisfaction of retailers has also been considered. For this purpose, a mixed integer programing model is proposed to design a cross dock warehouse to minimize distances, minimize the vacant space of the warehouse, and maximize retailer’s satisfaction.
Keywords: Multi-objective gray wolf optimization algorithm, Cross dock warehouse, Integer programming, Taguchi, Design, Layout -
Journal of Optimization in Industrial Engineering, Volume:15 Issue: 32, Winter and Spring 2022, PP 57 -66Ethiopia's industrial development strategy is characterized by manufacturing-led and expansion labor-intensive industrialization. The country expects to generate more income from the exported market. However, the case company is still known not to become productive as much as possible due to different reasons. One of the big challenges of the company has the problem with holding inappropriate inventory and with determines their optimal cost due to poor production planning. So that to solve this problem objective of the paper is to minimize total cost through the integration of seasonal forecasting and integer programming model without violating demand fulfillments. This technique improves resource utilization and enhances inventory control or stock control system. Currently, the company produces different kinds of products grouped into four common types of products (knitted garment, knitted fabric, woven garment, and woven fabric). The data survey system was both primary and secondary system and classified the products using A B C (always better classification) classification. The optimal solution was settled through the integration of seasonal forecasting and integer programming. As the Sensitivity analysis indicated the a big gap between production capacity and actual demand of the products. As the optimized solution indicated that total cost of production cost and inventory cost was minimized and the optimal production plan as well safety stock levels in each quarter was settled. Seasonal demand forecasting is a key activity for a garment which more or less controls all activities of production processes since garment products are affected by seasonal. As the result and discussion have shown that after optimized increase profit of the company through minimizing production cost and inventory costs since both costs are the big constraint of the company. Based on the optimized solution finding annually total cost needs for each A, B, and C – categories products are 57,225,920 BIRR 4,733,013 BIRR, 8,229,309 BIRR, respectively for production and inventory costs. The optimized solution indicated that if the company implemented exactly the proposed solution it will get an additional,4,219,788.8 BIRR,772,055.8 BIRR,2,119,824.2 BIRR respectively for A, B, C categories products totally around 7,111,668.8 BIRR profit per year will get. To end, it was concluded that this remarkable profit increment of the case company can certainly enhance its productivity and worldwide competitiveness. This research will create further pathways for other researchers to accomplish substantial studies on other garment sectors or other manufacturing industries based on local and international perspectives.Keywords: ABC classification, seasonal forecasting, Optimization, Integer programming
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Undergraduate thesis examination in Industrial Engineering Department of Universitas Sebelas Maret conducted through two stages, namely intermediate and final examination. Currently, the scheduling process of such examinations is done by the undergraduate thesis coordinator manually without certain systematic method or approach. In this paper, we develop an optimization model for the examinations scheduling considering several factors, namely the number of lecturers that must attend the examinations, the availability of rooms for examinations, the availability of each lecturer, and the assignment distributions. The model uses integer programming approach. Two performance criteria are used in the model, namely the difference between the number of each lecturer’s assignment with the average number of lecturer assignments and the number of penalties from the assignment of lecturers on certain time slot. The developed model is able to solve the scheduling problem more efficiently than manual scheduling done by thesis coordinator. The optimal solutions from the optimization model show a total difference in the assignment of lecturer with an average of 29.6 and a penalty of 0.
Keywords: Scheduling, Timetabling, Integer programming, Invigilator assignment -
International Journal of Research in Industrial Engineering, Volume:8 Issue: 3, Summer 2019, PP 243 -253Food product has a characteristic of continuous quality deterioration until the food is consumed. Cold chain distribution can improve the sustainability of the product quality but requires a more significant investment for storage facilities and vehicles as well as higher operation cost to control the temperature. This research focuses on a distribution problem faced by an ice cream distributor. In this paper, we developed a mixed-integer non-linear programming model to minimize the total cost, which consists of fixed cost, transportation cost of the vehicles, energy cost to keep the cold storage temperature, and inventory cost. The model considers the vehicle characteristics and hard time-windows for the distributor and all the stores. The implementation of this model demonstrates that the proposed route is able to reduce the total cost.Keywords: Frozen food, cold chain, perishable, Distribution, Integer programming, Time-Windows
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Journal of Optimization in Industrial Engineering, Volume:12 Issue: 25, Winter and Spring 2019, PP 173 -194During the last decade, reverse logistics networks received a considerable attention due to economic importance and environmental regulations and customer awareness. Integration of leading and reverse logistics networks during logistical network design is one of the most important factors in supply chain. In this research, an Integer Linear Programming model is presented to design a multi-layer reverse-leading, multi-product, and multi-period integrated logistics network by considering multi-capacity level for facilities under uncertainty condition. This model included threeobjectivesmaximizing profit, minimizing delay of goods delivering to customer, and minimizing returned raw material from suppliers. This research gives financial incentives to encourage customers in order to return their used product. Considering that the remaining value of used products is the main incentive of a company to buy second-handed goods, a dynamic pricing approach is determined to define purchase price for these types of products, and based on that, the percentage of returned products were collected by customers. In addition, in this study, parameters have uncertainty features and are vague; therefore, at first, they are converted into exact parameters and, then, because model is multi-objective, the fuzzy mathematical programming approach is used to convert multi-objective model into a single objective; finally, the model by version 8 of Lingo is run. In order to solve a large-sized model, a non-dominated sorting genetic algorithm II (NSGA-II) was applied. Computational results indicate the effect of the proposed purchase price on encourage customers to return the used products.Keywords: Integrated supply chain network, Fuzzy mathematical programming, Dynamic pricing approach, Integer programming, Quality levels
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International Journal of Research in Industrial Engineering, Volume:7 Issue: 1, Spring 2018, PP 1 -18
Effective design and management of Supply Chain Networks (SCN) support the production and delivery of products at low cost, high quality, high variety, and short lead times. In this study, a SCN is designed for an automotive company by integrating various approaches. The study has been carried out in two phases: The first phase involves selecting suppliers and distributors by using Data Envelopment Analysis (DEA) and integer-programming model. In the second phase, first the priority ranking of selected suppliers and distributors is determined using the Analytical Hierarchy Process (AHP) and then these priority rankings are integrated into the transportation models developed to identify the optimal routing decisions for all members of the supply chain.
Keywords: Supply chain management, Data Envelopment Analysis, Integer programming, Analytical Hierarchy Process, Transportation problem -
This paper deals with the determination of machine numbers and production schedules in manufacturing environments. In this line, a two-stage fuzzy stochastic programming model is discussed with fuzzy processing times where both deterioration and learning effects are evaluated simultaneously. The first stage focuses on the type and number of machines in order to minimize the total costs associated with the machine purchase. Based on the made decisions, the second stage aims to schedule orders, while the objective is to minimize total tardiness costs. A dependent-chance programming (DCP) approach is used for the defuzzification of the proposed model. As the resulted formulation is a NP-hard problem, a branch and bound (B&B) algorithm with effective lower bound is developed. Moreover, a genetic algorithm (GA) is proposed to solve problems of large-sizes. The computational results reveal the high efficiency of the proposed methods, in particular the GA, to solve problems of large sizes.Keywords: Scheduling, Design of production systems, Fuzzy methods, Integer programming, learning, Meta-heuristics
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امروزه با رشد سریع تکنولوژی، جمع آوری اطلاعات زیاد و مورد نیاز برای تجزیه و تحلیل فرایند در سازمان ها ممکن شده است، اما مهم تراز آن امکان به کارگیری بهینه و موثر این تعداد زیاد داده ها در تجارت مدرن است. هدف این مقاله استفاده از سیستم ماهالانوبیس تاگوچی برای حذف اطلاعات اضافی، و نیز تجزیه و تحلیل و تشخیص آنها سیستم های چندبعدی است. سیستم ماهالانوبیس تاگوچی از دو بخش کلی تشکیل شده است: 1. انتخاب متغیرهای مفید؛ 2. پیش بینی و تشخیص. در این سیستم برای غربالگری متغیرها از آرایه های متعامد تاگوچی استفاده شده است. در حقیقت نوآوری اصلی این مقاله، استفاده از مفهوم «طبقه بندی اشتباه و مدل سازی برنامه ریزی ریاضی عدد صحیح» به جای آرایه های متعامد تاگوچی به منظور به دست آوردن مجموعه یی مفید از متغیرهاست. با این روش دست یابی به مجموعه یی بهتر در زمانی کوتاه تر ممکن می شود.کلید واژگان: سیستم های چندمتغیره، سیستم ماهالانوبیس تاگوچی، برنامه ریزی عدد صحیحFor many years, analysis of real systems has attracted much attention. Such systems are hard to describe because of their complex behavior and their huge number of parameters and mutual effects. This issue has made analysis methods of a multivariable system develop rapidly. Multivariate analysis consists of a collection of methods that can be used when several measurements are made of each individual or object in one or more samples. In practice, multivariate data sets are common, although they are not always analyzed in the same way. However, the exclusive use of univariate procedures with such data is no longer acceptable, given the availability of multivariate techniques and inexpensive computing power. The Mahalanobis-Taguchi system is such a novel method. Nowadays, rapid development of technology has made it possible for organizations to gather large amounts of data for analyzing processes. But, on the other hand, an appropriate approach for dealing with these huge amounts is also required in modern commerce. The main purpose of this paper is to use Mahalanobis Taguchi systems for effective selection after reorganizing and omitting non-meaning data. This method is for multivariable systems and consists of two parts: The first part deals with a useful variable selection for complexity reduction and the second part contains recognition and prediction processes and identification of abnormal groups. The conventional method uses orthogonal Taguchi arrays for variable selection. In fact, the main novelty of this paper is the use of concepts such as mis classification and Integer Programming. The solution in our method is based on a creative algorithm, which is also accurate enough. It will be shown that this method is faster than former methods, and that its performance is generally better than previous methods.Keywords: Multivariable systems, mahalanobis, taguchi system, integer programming
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Journal of Industrial Engineering and Management Studies, Volume:2 Issue: 2, Summer-Autumn 2015, PP 55 -82In todays competitive market place, companies seek an efficient structure of supply chain so as to provide customers with highest value and achieve competitive advantage. This requires a broader perspective than just the borders of an individual company during a supply chain. This paper investigates an aggregate production planning problem integrated with distribution issues in a supply chain so as to simultaneously optimize characteristics of these supply chain drivers. The main contribution of this paper is to consider the aggregate production-distribution planning (APDP) problem jointly with multiple stage, multiple product, and multiple vehicle. Moreover, we considered both routing and direct shipment as transportation system which is not considered in APDP literature so far. A mixed-integer linear programming formulation is suggested for two distinct Scenarios: (i) when we have direct shipment in which all shipments are transported directly from manufacturer to customers, and (ii) when we have routing option in which the vehicles can move through routes to deliver products to more than one customer at a trip. A numerical analysis is performed to compare performance of problem in two above Scenarios. Moreover, to assess applicability of problem, some computational experiments are implemented on small, medium and large sized problems.Keywords: Mixed, Integer Programming, Production Planning, Production, Distribution, Transportation, Vehicle Routing, Setup Times
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جدول زمان بندی دروس دوره های دانشگاهی یک مسئله ی بزرگ تخصیص منابع است، که در آن هم زمان و هم مکان برای هر جلسه ی کلاس تعیین می شوند.که اکثر ادبیات های علمی راه حل کاملا اکتشافی را مطرح می کنند. برخی از روش ها در مورد برنامه ریزی زمان بندی کلاس ها مطرح شده اند اما این روش ها تنها برای دانشگاه های کوچک یا یک بخش و گروه در دانشگاه بزرگ تر مطرح شده اند، برای دانشگاه های بزرگ باید تجزیه ای بین بخش های آن صورت بگیرد و به بخش های کوچک تر تقسیم شود سپس آن را حل کنند که هم هزینه بر و هم زمان بر است و برای دانشگاه های بزرگ مناسب نیست. در این پژوهش به زمان بندی امتحانات با در نظر گرفتن اطلاعات( کلاس ها و امتحانات و ظرفیت کلاس ها و تعداد شرکت کنندگان امتحانات و ویژگی های خاص کلاس ها و امتحانات) به بررسی زمان بندی امتحانات در دانشگاه آزاد اسلامی قزوین پرداخته شد. این پژوهش کاربردی و ازنظر روش کمی-ریاضی است و برای همه دانشگاه ها و موسسات کوچک و بزرگ برای یافتن جواب بهینه برای تخصیص امتحانات و دروس به کلاس ها به صورت مستقیم و بدون نیاز به تجزیه بخش ها قابل استفاده است. همچنین این روش برای مسائل بزرگ منعطف است و در صرف زمان و هزینه صرفه جویی می شود. برای این منظور ابتدا مسئله با استفاده از الگوی برنامه ریزی عدد صحیح[1] تعریف شد و سپس با استفاده از الگوریتم فرا ابتکاری ژنتیک[2] حل می شود. خروجی محاسبات عدد دودویی بود که تعلق امتحان به کلاس را تعریف می کرد به طوری که هزینه تخصیص نادرست به صفر نزدیک شد.کلید واژگان: برنامه ریزی عدد صحیح، تخصیص کلاس درس، مقیاس بزرگIn this paper we present an integer programming method for solving the Classroom Assignment Problem in University CourseTime tabling.We introduce a novel formulation of the problem which generalizes existing models and maintain stractability even for large instances. The model isvalidated through computation a lresults base do nourexperiencesat the University of Auckland,and on instances from the 2007 International Timetabling Competition.We also expand uponexistingresultsin to the computational difficulty of roomassignment problemsKeywords: Integer programming, Classroom Assigment problem, Large-scale problem
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در این مطالعه یک مدل برنامه ریزی خطی عدد صحیح مختلط برای تصمیمات مکان یابیٓ تخصیص در یک شبکه ی زنجیره ی تامین چهارسطحی بیان شده است. اولین سطح این شبکه ی زنجیره ی تامین به تامین کنندگان، دومین سطح به کارخانه ها، سومین سطح به توزیع کنندگان و آخرین سطح آن به مناطق مشتریان اختصاص دارد. در این مدل هدف بیشینه کردن ارزش خالص فعلی جریان نقدی شبکه ی زنجیره ی تامین است. در این شبکه هزینه ی احداث کارخانه ها توسط سرمایه ی اولیه ی موجود یا با وام گرفتن از بانک به دست می آید. مدل ریاضی مکان یابیٓ تخصیص شامل هزینه ی سرمایه گذاری، هزینه ی تهیه ی ماده ی اولیه، هزینه ی تولید کالا، هزینه ی احداث، هزینه ی نگه داری موجودی و هزینه ی ارسال کالاهاست. در مدل ارائه شده، ارزش خالص فعلی جریان نقدی بعد از کسر مالیات برای دوره های زمانی مختلف بیشینه می شود. در نهایت، در این مطالعه برای توضیح هرچه بهتر مدل ارائه شده یک مثال عددی با نرم افزار G A M S حل شده است.
کلید واژگان: مسائل مکان یابی، تخصیص، زنجیره ی تامین، ارزش خالصفعلی (N P V)، برنامه ریزی عدد صحیح مختلطSupply chain management is a key issue and practical for sustainable economy, where its economic advantages can be shown in many papers. In order to remain competitive in the market, a supply chain must be efficient. One way to improve the efficiency is to maximize the total profit in the system. In this study, a mix-integer linear programming model is presented for location-allocation decisions in a four-echelon supply chain network. The first to the fourth echelon of the supply chain network includes suppliers, production plants, distribution centers, and customer zones, respectively. This paper amis to select the appropriate amount of loan and to adopt proper location-allocation decisions in all echelons of the chain in several periods, while maximizing the net present value. the net present value. Location-allocation decisions include locating a number of plants among a finite set of potential sites and decisions relate to the flow of goods from suppliers to production plants, from plants to distribution centers and from distribution centers to customer zones in each period. The suppliers prepare a similar raw material and the production plants produce a similar product from the raw material. The mix integer linear mathematical model of the location-allocation problem includes sale price, investment costs, raw material costs, production costs, setup costs, holding costs, shortage costs and transportation cost of goods. Setup costs of the factories involved in the network are obtained by an initial capital or funds borrowed from banks. Also, shortages are allowed and are considered to be lost sale. The proposed model maximizes the net present value of the profit after tax in different time periods using continuous compound interest rate. In order to demonstrate the applicability of the proposed formulation and to validate the results obtained, a numerical example is solved at the end using GAMS software. Finally, conclusion and future research recommendations come in the last section of paper.Keywords: Location, allocation problems, supply chain, net present value (NPV), mix, integer programming -
Journal of Industrial Engineering and Management Studies, Volume:2 Issue: 1, Winter-Spring 2015, PP 61 -73In this paper, a new mathematical model for the problem of job scheduling in virtual manufacturing cells (VMC) is presented to minimizing the completion time of all jobs. Sequence dependent setup times of machines is considered and lot-streaming is possible. In Virtual manufacturing cells, each job has a different processing path and there is a set of machines for processing each operation. There are multiple machine types with several identical machines in each type locating in different locations in the shop floor. In this type of system, the cells are not physical and Machines can be shared between the cells. In Mixed-integer nonlinear programming model presented, the scheduling decisions involve assigning a machine to each operation, the start time at each operation, the start time of machines and sub-lot sizes of each job. Some test problems have been generated to demonstrate the implementation of the model and solved by Lingo.Keywords: virtual manufacturing cells, Scheduling, lot, streaming, Mathematical model, integer programming
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Cancer is one of the major causes of death all over the globe and radiotherapy is considered one of its most effective treatment methods. Designing a radiotherapy treatment plan was done entirely manually in the past. RecentlyIntensity Modulated Radiation Therapy (IMRT) was introduced as a new technology with advanced medical equipmentin the recent years. IMRT provides the opportunity to deliver complex dose distributions to cancer cells while sparing the vital tissues and cells from the harmful effects of the radiations. Designing an IMRT treatment plan is a very complex matter due to the numerous calculations and parameters which must be decided for. Such treatment plan is designed in three separate phases: 1) selecting the number and the angle of the beams 2) extracting the intensity matrix or the corresponding dose map of each beam and 3) realizing each intensity matrix. The third phase has been studied in this research and a nonlinear mathematical model has been proposed for multileaf collimators. The proposed model has been linearized through two methods and an algorithm has been developed on its basis in order to solve the model with cardinality objective function. Obtained results are then compared with similar studies in the literature which reveals the capability of proposed method.Keywords: Intensity modulated radiation therapy (IMRT), Decomposing intensity matrix, Multileaf collimator, Benders decomposition, Integer programming
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افزایش چشم گیر جابه جایی مواد خطرناک نگراتی های را در مورد صدمات به انسان و طبیعت به وجود آورده است، بنابراین کاهش ریسک به عنوان معیاری مهم در محاسبات در نظر گرفته می شود. به طور خاص در این مقاله یک مدل دو هدفه خطی مختلط عدد صحیح برای مسئله توزیع مواد سوختی در شبکه با انبار توزیع چندگانه ارائه می گردد، به نحوی که مجموع هزینه انتقال و تاسیس انبار و مجموع ریسک برای انتقال مواد سوختی از انبار توزیع به نقاط تقاضا حداقل می شود. برای صحه گذاری مدل ریاضی، یک مثال عددی حل می شود و نتایج حاصل ارائه می گردد.
کلید واژگان: مکان یابی، مسیریابی، مواد خطرناک، مدل دو هدفه، مختلط عدد صحیحSince hazardous material transportation has been dramatically increased in terms of quantity, their incident impacts are getting to be known as a serious concern. Therefore, transport risk is usually considered as a main factor in hazardous material transportation and management. In this paper, a bi-objective mathematical model is developed for solving the fuel distribution problem, in which routing and location problems are solved simultaneously. While the problem is formulated as a mix-integer programming (MIP) model, distribution centers are considered as candidates, in which its main aims are to select the best set of establishing centers and find the best routes for carrying fuels from selected distribution centers to gas stations. The proposed model is eventually validated using the experimental data. Finally, the associated results are analyzed and the conclusion is provided.Keywords: Locating, routing problem, Bi, objective model, Hazardous material, Mixed, integer programming -
International Journal of Industrial Engineering and Productional Research, Volume:24 Issue: 3, Sep 2013, PP 217 -227Location-allocation of facilities in service systems is an essential factor of their performance. One of the considerable situations which less addressed in the relevant literature is to balance service among customers in addition to minimize location-allocation costs. This is an important issue, especially in the public sector. Reviewing the recent researches in this field shows that most of them allocated demand customer to the closest facility. While, using probability rules to predict customer behavior when they select the desired facility is more appropriate. In this research, equitable facility location problem based on the gravity rule was investigated. The objective function has been defined as a combination of balancing and cost minimization, keeping in mind some system constraints. To estimate demand volume among facilities, utility function(attraction function) added to model as one constraint. The research problem is modeled as one mixed integer linear programming. Due to the model complexity, two heuristic and genetic algorithms have been developed and compared by exact solutions of small dimension problems. The results of numerical examples show the heuristic approach effectiveness with good-quality solutions in reasonable run time.Keywords: Facility location, Equitable load, Gravity model, Heuristic algorithm, Genetic algorithm, Integer programming
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در این مقاله یک مدل ریاضی و یک روش ابتکاری جدید برای حل مسئله مسیریابی وسایل نقلیه چند قرارگاهی و چند محصولی با وسایل نقلیه متفاوت ارائه شده است. مشتریان امکان سفارش کالاهای متفاوت را دارند، قرارگاه ها نیز باید با وسایل نقلیه ای که دارای ظرفیت، سرعت و هزینه های مختلفی هستند، به سفارشات مشتریان خود قبل از موعد تحویل پاسخ دهند. به همین منظور مدل ریاضی مسئله مسیریابی چند قرارگاهی توسعه یافته است تا بتواند بازتاب دهنده شرایط مطرح شده باشد. هدف این مدل کاهش مسافت سرویس دهی است، که منجر به کاهش در هزینه ها و زمان سرویس دهی می شود. به دلیل پیچیدگی زیاد این مسئله و زمان بر یا غیر ممکن بودن حل دقیق مدل توسعه یافته، یک روش ابتکاری پیشنهاد شده است. این روش دارای چهار گام اصلی شامل: گروهبندی مشتریان، مسیریابی و تعیین نوع وسیله نقلیه، زمانبندی و بسته بندی سفارشات و بهبود است. کارایی این روش با مقایسه جواب های بهینه چندین مسئله و یک مطالعه موردی در شرکت کاله یزد مورد بررسی قرار گرفت. نتایج حاصل، موثر بودن این روش و قابلیت استفاده از آن در صنعت را نشان داد.کلید واژگان: مسیریابی وسایل نقلیه چند قرارگاهی، برنامه ریزی عدد صحیح، روش ابتکاری، حداقل کردن هزینهInternational Journal of Industrial Engineering & Production Management, Volume:23 Issue: 3, 2012, PP 303 -315A mathematical model and heuristic method for solving multi-depot and multi-product vehicle routing problem with heterogeneous vehicle have been proposed in this paper. Customers can order several products and depots must deliver customer's orders before due date with different vehicle. Hence mathematical model of multi-depot vehicle routing problem has been developed to represent these conditions. Aim of this model is to minimize total delivery distance or time spent in servicing all customers. As this problem is very complex, we have offered a heuristic method that includes four steps. Grouping, routing and vehicle selection, scheduling and packing of products and improvement are the aforementioned steps. Efficiency of heuristic has been tested by a case study and several numerical examples. Comparing the results of heuristic and optimal solving has revealed that the deviation of heuristic results from optimal answer is lower than eight perce.Keywords: Multi, Depot Vehicle Routing Problem, Integer Programming, Heuristic Method, Minimize cost
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در این مقاله یک مفهوم جدید در حوزه مکان یابی، به نام «مسئله مکان یابی با بار کاری متوازن بر مبنای مدل جاذبه» معرفی شده است. متوازن سازی در جستجوی توزیع عادلانه تقاضا یا برقراری تعادل در ظرفیت تسهیلات هنگام پاسخگویی به تقاضای مشتریان است؛ به طوری که آنها تسهیلات را بر اساس معیار مناسب تری مانند قانون جاذبه انتخاب کنند. تابع هدف در صدد کمینه سازی حداکثر بار کاری موجود بین تسهیلات، هزینه استقرار آنها و هزینه جابه جایی تا مشتریان، با در نظر گرفتن مدل جاذبه است. در ادامه، بر اساس ساختار مسئله، یک رویکرد حل ابتکاری طراحی و روی مثال های عددی با ابعاد مناسب با روش دقیق، تحلیل مقایسه ای انجام شده است. نتایج محاسباتی نشان می دهد که رویکرد حل پیشنهادی، با میانگین اختلاف تقریبا 6درصد و زمان محاسباتی قابل قبول، روشی بسیار کارآ بوده و جواب های نزدیک به بهینه را برای مثال های تصادفی تولیدشده ایجاد می کند.
کلید واژگان: متعادل سازی، مکان یابی تسهیلات، برنامه ریزی عدد صحیح، بار متوازن، الگوریتم ابتکاری، قانون جاذبهThis paper introduces a new concept in location problems, called "Gravity-Based Equitable Location Problem". Equity is seeking for a fair distribution of demands or balancing the facility capacity to meet demands, so that the customers select them by appropriate criteria like the gravity law. The objective function defined as minimizing the maximum of facilities workload and deployment and movement costs regarding to the gravity model. Then, a heuristic algorithm designed by a problem structure and comparative analysis accomplished with the exact method on appropriate numerical examples. The proposed algorithm yields near optimal solutions for randomly generated examples. Results show that it is very efficient, with the mean difference of approximately 6 percent and acceptable computational timeKeywords: Gravity model, Equitable load, Balancing, Integer programming, Facility location -
Presented here is a generalization of the implicit enumeration algorithm that can be applied when the objective function is being maximized and can be rewritten as the difference of two non-decreasing functions. Also developed is a computational algorithm, named linear speedup, to use whatever explicit linear constraints are present to speedup the search for a solution. The method is easy to understand and implement, yet very effective in dealing with many integer programming problems, including knapsack problems, reliability optimization, and spare allocation problems. To see some application of the generalized algorithm, we notice that the branch-and-bound is the popular method to solve integer linear programming problems. But branch-andbound cannot efficiently solve all integer linear programming problems. For example, De Loera et al. in their 2005 paper discuss some knapsack problems that CPLEX cannot solve in hours. We use our generalized algorithm to find a global or near global optimal solutions for those problems, in less than 100 seconds. The algorithm is based on function values only; it does not require continuity or differentiability of the problem functions. This allows its use on problems whose functions cannot be expressed in closed algebraic form. The reliability and efficiency of the proposed algorithm has been demonstrated on some integer optimization problems taken from the literature.
Keywords: Algebraic form, Function values, Generalized implicit enumeration, Integer programming, Linear speedup
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